Determinant of Laplacians on Heisenberg manifolds
We give an integral representation of the zeta-regularized determinant of Laplacians on three-dimensional Heisenberg manifolds, and study a behavior of the values when we deform the uniform discrete subgroups. Heisenberg manifolds are the total space of a fiber bundle with a torus as the base space...
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Published in | Journal of geometry and physics Vol. 48; no. 2; pp. 438 - 479 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier B.V
01.11.2003
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Abstract | We give an integral representation of the
zeta-regularized determinant of Laplacians on three-dimensional Heisenberg manifolds, and study a behavior of the values when we deform the uniform discrete subgroups. Heisenberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the “radius” of the fiber goes to zero. We explain the lines of the calculations precisely for three-dimensional cases and state the corresponding results for five-dimensional Heisenberg manifolds. We see that the values themselves are of the product form with a factor which is that of the flat torus. So in the last half of this paper we derive general formulas of the zeta-regularized determinant for product type manifolds of two Riemannian manifolds, discuss the formulas for flat tori and explain a relation of the formula for the two-dimensional flat torus and the
Kronecker’s second limit formula. |
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AbstractList | We give an integral representation of the
zeta-regularized determinant of Laplacians on three-dimensional Heisenberg manifolds, and study a behavior of the values when we deform the uniform discrete subgroups. Heisenberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the “radius” of the fiber goes to zero. We explain the lines of the calculations precisely for three-dimensional cases and state the corresponding results for five-dimensional Heisenberg manifolds. We see that the values themselves are of the product form with a factor which is that of the flat torus. So in the last half of this paper we derive general formulas of the zeta-regularized determinant for product type manifolds of two Riemannian manifolds, discuss the formulas for flat tori and explain a relation of the formula for the two-dimensional flat torus and the
Kronecker’s second limit formula. |
Author | Furutani, Kenro de Gosson, Serge |
Author_xml | – sequence: 1 givenname: Kenro surname: Furutani fullname: Furutani, Kenro email: furutani_kenro@ma.noda.tus.ac.jp organization: Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, 2641 Noda, Chiba 278-8510, Japan – sequence: 2 givenname: Serge surname: de Gosson fullname: de Gosson, Serge email: sdg@vax.se organization: Department of Mathematics, Växjö University, SE-351 95 Växjö, Sweden |
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CitedBy_id | crossref_primary_10_1007_s00022_010_0034_2 crossref_primary_10_1016_j_aim_2014_11_017 crossref_primary_10_1016_j_geomphys_2008_07_011 crossref_primary_10_1016_j_geomphys_2010_04_009 crossref_primary_10_1016_j_matpur_2011_06_003 crossref_primary_10_1016_j_geomphys_2007_09_007 crossref_primary_10_1063_1_4936074 |
Cites_doi | 10.1007/BF01391828 10.1103/RevModPhys.60.917 10.1090/S0002-9947-71-99991-0 10.1016/0001-8708(71)90045-4 10.1307/mmj/1029003354 10.1002/cpa.3160250302 10.1063/1.531134 10.1137/0519035 10.1017/CBO9781107325937 10.3792/pja/1195521077 10.1080/03605309608821191 10.1007/BF02096935 10.2307/2154453 |
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Keywords | Heisenberg group Zeta-regularized determinant Modified Bessel function Poisson’s summation formula Heat kernel Kronecker’s second limit formula Laplacian |
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References | Vardi (BIB14) 1988; 19 Berndt (BIB2) 1971; 160 Mckean (BIB9) 1972; 25 Motohashi (BIB10) 1968; 44 Furutani (BIB8) 1996; 21 Quine, Heydari, Song (BIB12) 1993; 338 Gordon, Wilson (BIB6) 1986; 33 Ray, Singer (BIB13) 1971; 7 Berndt (BIB3) 1971; 160 G.E. Andrews, R. Askey, R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. D’Hoker, Phong (BIB5) 1988; 60 Bolte, Steiner (BIB4) 1990; 130 Forman (BIB7) 1987; 88 Nash, O’Connor (BIB11) 1995; 36 Berndt (10.1016/S0393-0440(03)00053-6_BIB3) 1971; 160 Furutani (10.1016/S0393-0440(03)00053-6_BIB8) 1996; 21 Ray (10.1016/S0393-0440(03)00053-6_BIB13) 1971; 7 Berndt (10.1016/S0393-0440(03)00053-6_BIB2) 1971; 160 Bolte (10.1016/S0393-0440(03)00053-6_BIB4) 1990; 130 D’Hoker (10.1016/S0393-0440(03)00053-6_BIB5) 1988; 60 Gordon (10.1016/S0393-0440(03)00053-6_BIB6) 1986; 33 Motohashi (10.1016/S0393-0440(03)00053-6_BIB10) 1968; 44 Mckean (10.1016/S0393-0440(03)00053-6_BIB9) 1972; 25 Vardi (10.1016/S0393-0440(03)00053-6_BIB14) 1988; 19 Forman (10.1016/S0393-0440(03)00053-6_BIB7) 1987; 88 Nash (10.1016/S0393-0440(03)00053-6_BIB11) 1995; 36 Quine (10.1016/S0393-0440(03)00053-6_BIB12) 1993; 338 10.1016/S0393-0440(03)00053-6_BIB1 |
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Snippet | We give an integral representation of the
zeta-regularized determinant of Laplacians on three-dimensional Heisenberg manifolds, and study a behavior of the... |
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StartPage | 438 |
SubjectTerms | Heat kernel Heisenberg group Kronecker’s second limit formula Laplacian Modified Bessel function Poisson’s summation formula Zeta-regularized determinant |
Title | Determinant of Laplacians on Heisenberg manifolds |
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